​In this paper, we investigate Hyers–Ulam and Hyers–Ulam–Rassias stability for fractional linear Hahn difference equations of Caputo Type. To the best of our knowledge, this is the first work concerning Hyers–Ulam type stability in the framework of Caputo fractional Hahn difference equations, thereby filling a significant gap in the literature on fractional stability theory. Our approach is based on establishing an equivalence between the fractional Hahn difference initial value problem and a corresponding fractional integral equation, via the Caputo–Hahn inversion formula. These results lay a strong groundwork for analyzing the stability of Hahn fractional systems, which could be useful in various scientific and engineering fields. Finally, illustrative examples are given to show the applicability of the theoretical results.

Motivated by some recent results concerning the stability of second-order systems of nonlinear difference equations, we aim in this paper to investigate the global asymptotic stability of a third-order twodimensional system. Furthermore, we discuss the convergence of solutions of this system. Moreover, we establish two asymptotic relations for solutions. Finally, many illustrative examples are given

In this paper, we obtain new sufficient conditions for the existence and uniqueness of solutions of Caputo fractional linear differential equations, via Banach fixed point theorem. Then, we investigate Hyers-Ulam and Hyers-Ulam-Rassias stability for such equations. Finally, two illustrative examples are given to show the applicability of the theoretical results.

In this paper, we investigate Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear quantum difference equations associated with a general quantum difference operator. This operator includes as special cases Jackson q-difference and Hahn difference operators. At the end of the paper, an illustrative example is given to show the applicability of the theoretical results.

To assess the feasibility and outcomes of immediate breast reconstruction using a dermal sling and polypropylene mesh for fixation after skin-reducing mastectomy with prepectoral prosthesis placement in patients with large breasts who were diagnosed with cancer. This retrospective study included demographic and clinical data from female patients with breast cancer and large breasts, who were candidates for skin-reducing mastectomy and immediate reconstruction. Data regarding operative technique, implant size, operative duration, and intraoperative complications were retrieved. Early and late postoperative complications were recorded. Routine assessment of postoperative patient satisfaction was performed using the Breast Reconstruction Satisfaction Questionnaire. The study included data from 49 female patients with a mean (± SD) age of 40.88 ± 8.03 years; the mean follow-up was 31.5 ± 8.1 months. The operation was successful in 47 (95.9 www.springer.com/00266 .

In this paper, we revisit the L p -spaces, p 1, associated with a general quantum difference operator and prove some convergence theorems in the quantum setting.Furthermore, two inequalities of Hardy's type are established.Finally, many illustrative examples concerning with q-difference operator, Hahn difference operator and power quantum difference operator are given.

Incidence of cervical lymph node metastasis to level V in patients withN1b PTC is low compared to levels II, III and IV.-There is clear evidence of postoperative morbidity from routine level V dissection in N1b PTC.-Level V dissection in patients with N1b PTC may be reserved for patients with clinically or radiologically evident level V metastasis. PATIENTS:The study included twenty patients with papillary thyroid cancer metastasizing to lateral cervical lymph nodes with no evidence clinically and radiologically of lymphadenopathy at level V.We excluded patients with previous thyroid or cervical lymph node surgery, patients with other head and neck malignancies and patients with history of neck irradiation.The study was conducted at Head, Neck and Endocrine surgery unit at Main Alexandria University Hospital.

Objective The aim was to assess involvement of level V cervical lymph nodes (LNs) in patients with stage N1b papillary thyroid carcinoma (PTC) and to determine the clinical risks and benefits of routine level V dissection in these patients. Patients and methods The study included 20 patients with papillary thyroid cancer metastasizing to cervical LNs with no evidence clinically or radiologically of lymphadenopathy at level V. All cases were managed by total thyroidectomy and modified radical neck dissection. The study was conducted at the head, neck, and endocrine surgery unit at Main Alexandria University Hospital, Alexandria, Egypt. Results Metastatic LNs were distributed in the different cervical levels according to postoperative histopathology as follows: level II LNs were positive for malignancy in 16 neck sides (80%), level III in 17 neck sides (85%), and level IV in 15 neck sides (75%). Level VI LNs were positive in 19 patients (95%). Level V was free of malignancy in all studied patients. Postoperative complications were as follows: shoulder dysfunction was noted in three patients (15%), neck numbness and neuralgia were noted in seven patients (35%), recent hoarseness of voice was noted in one patient (5%), one patient (5%) showed delayed extubation, and ear numbness was noted in five patients (25%). No patients in our study experienced postoperative hematoma, chyle leak, or manifestations of hypoparathyroidism. Conclusions Incidence of cervical LN metastasis to level V in patients with N1b PTC is low compared with levels II, III, and IV. Moreover, there is clear evidence of postoperative morbidity from routine level V dissection in these patients. Therefore, level V dissection in patients with N1b PTC may be reserved for patients with clinically or radiologically evident level V metastasis.

In this paper, we investigate the periodicity of two systems of rational sequences of second and third order, respectively.The systems include a permutation that gives the ability of changing the appearance of components of solutions in the equations of the systems.We find periods of systems in terms of the order of the permutation.The periodicity of two more systems of maximum type are studied.Finally, many illustrative examples are given.

Background: Chylous ascites (CA) is accumulation of lipid rich lymph in peritoneal cavity. CA is rare among children. In pediatric age, causes of CA varies according to the age group, leading primary causes include congenital malformation of lymphatic system, and less likely lymphatic obstruction such as intestinal malrotation, gastroschisis, infections or trauma of thoracic duct.  Aim of the work: We aim to report outcome of congenital CA in a pediatric cohort. Material and Methods: Retrospective analysis of data of 4 children (4 boys) who presented by CA to New Children Hospital, Cairo University during 2010- 2018.  Results: Duration of follow up of the boys was 6.5, 5, 2.5, and 0.25 years respectively. All presented by abdominal distention and diagnostic tapping of ascitic fluid revealed triglyceride level more than 200 mg%, protein more than 2 g% and/or a cell count greater than 500 /cmm with a predominance of lymphocytes (> 80%). Lymphatic scintigraphy in 3 of them revealed no abnormality in lymph drainage. The course of disease was punctuated by chylothorax, and chylous hydrocoele in all 4 cases. All 4 did not respond to medium chain triglyceride based diet. One child underwent peritenovenous shunt, and the other 3 responded to somatostatin analogue octreotide SC injections during hospital stay and maintenance therapy. Last child was lost to follow up. All other 3 are fine with minimal complications and living with mild peritoneal ascites.  Conclusion: Somatostatin analogue therapy provided well-tolerated non-invasive control of congenital CA in our studied pediatric cohort. Surgical intervention should be restricted to those with underlying surgical cause and in those failing to respond to medical treatment.

In this paper, we establish Taylor theory based on Hahn’s difference operator $D_{q,\omega}$ which is defined by $D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega}$, $t\neq\frac {\omega}{1-q}$, where $q\in(0,1)$ and ω is a positive number.

We will show in this paper that all solutions for the systems $ \varkappa _{n+1}^{(1)}=\frac{\varkappa _{n}^{(2)}}{\alpha \varkappa _{n}^{(2)}-1},\varkappa _{n+1}^{(2)}=\frac{\varkappa _{n}^{(3)}}{\alpha \varkappa _{n}^{(3)}-1},...,\varkappa _{n+1}^{(\kappa )}=\frac{\varkappa _{n}^{(1)}}{\alpha \varkappa _{n}^{(1)}-1},$ and $ \varkappa _{n+1}^{(1)}=\frac{\varkappa _{n}^{(\kappa )}}{\alpha \varkappa _{n}^{(\kappa )}-1},\varkappa _{n+1}^{(2)}=\frac{\varkappa _{n}^{(1)}}{ \alpha \varkappa _{n}^{(1)}-1},...,\varkappa _{n+1}^{(\kappa )}=\frac{ \varkappa _{n}^{(\kappa -1)}}{\alpha \varkappa _{n}^{(\kappa -1)}-1}, $ are periodic with period $p$ where $p$ is given by$p=\left\{ \begin{array}{c} \kappa \text{ if }\kappa =0(mod2), 2\kappa \text{ if }\kappa \neq 0(mod2), \end{array} \right\} $ where $\alpha $ and $\varkappa _{0}^{(1)},\varkappa _{0}^{(2)},...,\varkappa _{0}^{(\kappa )}$ are nonzero real numbers with $\varkappa _{0}^{(i)}\neq \frac{1}{\alpha },~i=1,2,...,\kappa $, for some $\kappa \in \mathbb{N}$.

In this paper, we study Hyers–Ulam stability and Hyers–Ulam–Rassias stability of first order non-linear impulsive time varying delay dynamic system on time scales, via a fixed point approach. We obtain some results of existence and uniqueness of solutions by using Picard operator. The main tools for our results are the Grönwall’s inequality on time scales, abstract Grönwall lemma and Banach contraction principle. In order to overcome difficulties arises in our considered model, we pose some conditions along with Lipchitz condition. At the end, an example is given that shows the validity of our main results.

In this paper, we investigate many types of stability, like (uniform stability, exponential stability and h-stability) of the first order dynamic equations of the form{u(Delta)(t) = Au(t) + f(t), t is an element of T, t > t(0) u(t(0)) - x is an element of D(A),and{u(Delta)(t) = Au(t) + f(t,u), t is an element of T, t > t(0) u(t(0)) = x is an element of D(A),in terms of the stability of the homogeneous equation(u(Delta)(t) = Au(t), t is an element of T, t > t(0) u(t(0)) = x is an element of D(A),where f is rd-continuous in t is an element of T and with values in a Banach space X, with f (t,0) = 0, and A is the generator of a C-0-semigroup {T(t) : t is an element of T} subset of L(X), the space of all bounded linear operators from X into itself. Here D(A) is the domain of A and T subset of R->= 0 is a time scale which is an additive semigroup with property that a - b is an element of T for any a, b is an element of T such that a > b. Finally, we give illustrative examples.

Hahn difference operator D-q,D-omega which is defined byD(q,omega)g(t) = {g(qt + omega)-g(t)/t(q - 1) vertical bar omega, if t not equal theta:= omega/1 - q, g'(theta), if t = thetareceived a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the formD(q,omega)x(t) = A(t)x(t) + f(t), t is an element of I,andD(q,omega)(2)x(t) + A(t) D(q,omega)x(t) + R(t)x(t) = f(t), t is an element of I,where A;R : I -> X, and f : I -> X. Here X is a Banach algebra with a unit element e and I is an interval of R containing theta.

In this paper we investigate sufficient conditions for many types of stability of both of the abstract first order linear dynamic equations on time scales of the formand the second order linear dynamic equations of the form ∆∆ + ∆ + = , ∊ , Where, : → (), the space of all bounded linear operators from a Banachspace into itself, and is rd-continuous from a time scale to .Some givenillustrative examples show the applicability of the main results.

Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture NotesFrontiers in Orthogonal Polynomials and q-Series, pp. 35-83 (2018) No AccessChapter 4: A Sturm–Liouville Theory for Hahn Difference OperatorM. H. Annaby, A. E. Hamza and S. D. MakhareshM. H. AnnabyDepartment of Mathematics, Faculty of Science Cairo University, P.O. Box 12613, Giza, Egypt, A. E. HamzaDepartment of Mathematics, Faculty of Science Cairo University, P.O. Box 12613, Giza, Egypt and S. D. MakhareshDepartment of Mathematics, Faculty of Science Cairo University, P.O. Box 12613, Giza, Egypthttps://doi.org/10.1142/9789813228887_0004Cited by:9 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: This chapter introduces a comprehensive study for Sturm–Liouville theory of the q, ω-Hahn difference operators in the regular setting. We define a Hilbert space of q, ω-square summable functions in terms of Jackson–Nörlund integral. The formulation of the self-adjoint operator and the properties of the eigenvalues and the eigenfunctions are discussed. The construction of Green's function is developed and a study for q, ω-Fredholem integral operator is established. Hence, an eigenfunctions expansion theorem is derived and illustrative examples are exhibited. We also introduce a separate section for numerical simulations and illustrations. We give some comparisons between trigonometric functions and the q and q, ω counterparts. We also test numerically the asymptotic behavior of the zeros of q and q, ω trigonometric functions. The numerical experiments precisely reflect the theoretical results with this respect. Dedication: Dedicated to Professor Mourad Ismail on the occasion of his 70th birthdayKeywords: Hahn difference operatorSturm–Liouville theoryq-difference operatorGreen's functioneigenfunctions expansionMSC 2010: 39A70, 39A12, 33D15 FiguresReferencesRelatedDetailsCited By 9On the spectrum of singular Hahn-Sturm-Liouville operatorsBilender P. Allahverdiev and Hüseyin Tuna1 Sep 2022 | Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15, No. 3On square integrable solutions of a Hahn–Dirac systemBilender P. Allahverdiev and Hüseyin Tuna28 August 2021 | Rendiconti del Circolo Matematico di Palermo Series 2, Vol. 43A β -Sturm–Liouville problem associated with the general quantum operatorJ. L. Cardoso20 May 2021 | Journal of Difference Equations and Applications, Vol. 27, No. 4Dirac System Associated with Hahn Difference OperatorFatma Hıra1 January 2020 | Bulletin of the Malaysian Mathematical Sciences Society, Vol. 43, No. 5Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville TypeBilender P. Allahverdiev and Hüseyin Tuna9 July 2020 | Communications in Mathematics, Vol. 28, No. 1A Representation of the Resolvent Operator of Singular Hahn-Sturm-Liouville ProblemBilender P. Allahverdiev and Hüseyin Tuna9 September 2019 | Numerical Functional Analysis and Optimization, Vol. 41, No. 4The Parseval Equality and Expansion Formula for Singular Hahn-Dirac SystemBilender P. Allahverdiev and Hüseyin Tuna1 Jan 2020A Completeness Theorem for a Hahn–Fourier System and an Associated Classical Sampling TheoremH. A. Hassan23 January 2019 | Results in Mathematics, Vol. 74, No. 1INDICES DEFECT THEORY OF SINGULAR HAHN-STURM-LIOUVILLE OPERATORSBilender P. Allahverdiev and Hüseyin Tuna1 Jan 2019 | Journal of Applied Analysis & Computation, Vol. 9, No. 5 Frontiers in Orthogonal Polynomials and q-SeriesMetrics History KeywordsHahn difference operatorSturm–Liouville theoryq-difference operatorGreen's functioneigenfunctions expansionPDF download

Let k ∈ N, Z k = {1, 2, . . ., k} and S k be the group of all permutations on Z k .Let π ∈ S k be of order l and fi be a function from a nonempty set X into itself, i = 1, . . ., k.In this paper, we show that a sufficient condition for a system of difference equationsx

Arithmetical calendar (or tabular calendar) is sometimes referred to as the Fātimid calendar but this is in fact one of several almost identical tabular Islamic calendars. This calendar introduced by Muslim astronomers in the 9th century CE to predict the approximate begin of the months in the Islamic lunar calendar. Chronologists adopted 11 leap years in a 30 year cycle. In the case of leap Hijri year they add one day to the last month of the Hijri year. The cycle of this calendar agree with the Smaller cycles (2–5.333 years) discovered by Galal and Rashed (2011) and coincide with the lag criterion given by Galal (1988).

In this paper, we use the Lyapunov's second method to obtain new sufficient conditions for many types of stability like exponential stability, uniform exponential stability, h-stability, and uniform h-stability of the nonlinear dynamic equation x(Delta)(t) = A(t)x(t) + f (t, x), t is an element of T-tau(+) := [tau, infinity)(T), on a time scale T, where A is an element of C-rd(T, L(X)) and f : T x X -> X is rd-continuous in the first argument with f(t, 0) = 0. Here X is a Banach space. We also establish sufficient conditions for the nonhomogeneous particular dynamic equation x(Delta)(t) = A(t)x(t) + f (t), t is an element of T-tau(+), to be uniformly exponentially stable or uniformly h-stable, where f is an element of C-rd(T, X), the space of rd-continuous functions from T to X. We construct a Lyapunov function and we make use of this function to obtain our stability results. Finally, we give illustrative examples to show the applicability of the theoretical results.